Chapter 4 Infinity in Greek Mathematics Fear of Infinity Eudoxus’ Theory of Proportion The Method of Exhaustion The Area of a Parabolic Segment Biographical Notes: Archimedes

4.1 Fear of Infinity Discovery of irrational numbers Greeks tried to avoid the use of irrationals The infinity was understood as potential for continuation of a process but not as actual infinity (static and completed) Examples: 1,2, 3,... but not the set {1,2,3,…} sequence x1, x2, x3,… but not the limit x = lim xn Paradoxes of Zeno (≈ 450 BCE): the Dichotomy there is no motion because that which is moved must arrive at the middle before it arrives at the end (cited from Aristotle’s “Physics”) Approximation of √2 by the sequence of rational number (Pell’s equation)

4.2 Eudoxus’ Theory of Proportions Eudoxus (around 400 – 350 BCE) The theory was designed to deal with (irrational) lengths using only rational numbers Length λ is determined by rational lengths less than and greater than λ Then λ1 = λ2 if for any rational r < λ1 we have r < λ2 and vice versa (similarly λ1 < λ2 if there is rational r < λ2 but r > λ1 ) Note: the theory of proportions can be used to define irrational numbers: Dedekind (1872) defined √2 as the pair of t wo sets of positive rationals L√2 = {r: r2< 2} and U√2 = {r: r2>2} (Dedekind cut)

4.3 The Method of Exhaustion was designed to find areas and volumes of complicated objects (circles, pyramids, spheres etc.) using approximations by simple objects (rectangles, trianlges, prisms) having known areas (or volumes) the Theory of Proportions

Examples Approximating the circle Approximating the pyramid

Example: Area of a Circle Let C(R) denote area of the circle of radius R We show that C(R) is proportional to R2 Inner polygons P1 < P2 < P3 <… Outer polygons Q1 > Q2 > Q3 >… Qi – Pi can be made arbitrary small Hence Pi approximate C(R) arbitrarily closely Elementary geometry shows that Pi is proportional to R2 . Therefore Pi(R) : Ri (R’) = R2:R’2 Suppose that C(R):C(R’) < R2:R’2 Then (since Pi approximates C(R)) we can find i such that Pi (R) : Pi (R’) < R2:R’2 which contradicts 5) P2 P1 Q1 Q2 Thus Pi(R) : Ri (R’) = R2:R’2

4.4 The area of a Parabolic Segment [Archimedes (287 – 212 BCE)] Triangles Δ1 , Δ2 , Δ3 , Δ4,… Note that Δ2 + Δ3 = 1/4 Δ1 Similarly Δ4 + Δ5 + Δ6 + Δ7 = 1/16 Δ1 and so on Y S Z 1 R 4 7 2 3 Q 6 5 O P X Thus A = Δ1 (1+1/4 + (1/4)2+…) = 4/3 Δ1

4.5 Biographical Notes: Archimedes Was born and worked in Syracuse (Greek city in Sicily) 287 BCE and died in 212 BCE Friend of King Hieron II “Eureka!” (discovery of hydrostatic law) Invented many mechanisms, some of which were used for the defence of Syracuse Other achievements in mechanics usually attributed to Archimedes (the law of the lever, center of mass, equilibrium, hydrostatic pressure) Used the method of exhaustions to show that that the volume of sphere is 2/3 that of the enveloping cylinder “Stay away from my diagram!”